A book about the history, development, and the ideas of chaos theory, by James Gleick.

Summary

Things actually began with something related to the butterfly effect, in climate modeling. Edward Lorenz was trying to predict weather patterns using computer-based models, and found (by accident) that the minor rounding errors in the intermediate computing steps resulted in vastly different final simulated result. He had stumbled across the world of chaos, in which, by his own words, “the present determines the future, but the approximate present does not approximately determine the future”.

Most people initially thought that chaos was simply random noise. So, like many other truly revolutionary ideas, it was met with rejection and ridicule from mainstream academia. But several people (including Lorenz himself, Mitchell Feigenbaum, and Albert Libchaber) soldiered on, did more research, and fully uncovered the world of chaos.

Chaos is intricately linked to dynamical systems and fractals. Dynamic systems are formed by the repeated evaluation of some relatively simple function, feeding the output from the previous iteration into the next one as input. For example, $f(x) = x^2 + 1$, when iterated repeated starting with $x = 0$, produces $1$, $2$, $5$, $26$, …

Chaos can emerge after certain iterations in dynamical systems with extremely simple functions, as in: minor variations in the initial seed value can produce vastly different iterated behaviors.

Fractal patterns can also emerge from certain very simple dynamic systems, when studying their behaviors. For example, we can examine a more general form of the above dynamical system, namely $f(x) = x^2 + c$. If we fix our initial seed for $x$ to be $0$, and instead ask the question for what values of $c$ (real or complex) does our dynamic system not diverge (i.e. not blow up to infinity, like the case above with $c = 1$ did), the resulting points forms the beautiful Mandelbrot set when plotted.

Once discovered, chaos was found to be everywhere in nature: in weather patterns, fluid dynamics, phase transitions, pendulums, animal population modeling, dripping faucets, and various fields of biology.

More than anything else, the discovery of chaos is significant as it is a new way to see the world that is vastly different from people’s traditional view before. Previously people thought that simple systems behave in simple ways, and that complex behavior implies complex causes. But with chaos this changed: simple systems can give rise extremely complex behaviors, and complex behaviors may have arisen from very simple systems (see the function that produced the Mandelbrot set above). Chaos shone the light on the nature of complexity itself, and provided a fresh way to look at old data. It marked the end of the purely reductionist era in science.

Commentary

The prevalence of chaotic behavior in nature is perhaps not so surprising, after realizing that it can emerge from dynamical systems. Dynamical systems, due to their repeated iteration process, really represent an abstract way to simulate certain mechanisms of reality. It may be crude, but it’s roughly how our world works from a high level.

Besides providing a brand new way to look at the world as the author has said, for me, chaos also paints a different picture of philosophical determinism. Previously, if we suppose that the world is fully deterministic, then the future is more or less fully predictable; even though our measurements cannot be infinitely precise, we can make precise-enough measurements to make accurate-enough predictions. Chaos changed this view completely: even if the world is fully deterministic from a theoretical sense, as long as infinitely precise measurements are impossible, we can never even approximately predict the future, because even minuscule errors can lead to vastly divergent results.

Even though I have only included the key technical concepts in the summary above, the book is filled with examples and details from the historical development of chaos as a theory, which are all quite interesting. One thing to keep in mind is that the accompanying diagrams in the book are key to better understand some of the core concepts, which listeners of the audio version of the book may not realize (I didn’t realized this myself until half way through the book).

If you don’t have the time to read through the book, this video by Veritasium covers many of the key ideas discussed in the book, animated. It is inspired by the book, but I found the video to be quite a bit more dense in information content than the book, per unit of time consumed.